Research on association schemes and related topics

首席研究员从以下三个方面进行了研究。自旋模型研究。(i)引入了4权自旋模型的概念，并证明了从中得到了环不变量（与Etsuko Bannai合作）。（ii）给出了有限阿贝尔群对偶性的完全分类和特征表上的模不变量，并证明了自旋模型是由这些解构造的（与Etsuko Bannai和F.Jaeger合著）。用码的权枚举数或有限群的多项式不变量构造各种自同构形式。(i)证明了192阶二维酉反射群（Shepherd-Todd's list中的No.9）的同时对角作用构成了Jacobi形式（与Michio Ozeki合作）。（ii）权重4的某些Jacobi形式的显式构造（与Michio Ozeki和Shinri Minashima合作）。（iii）我们确定了上述(i)中提到的多项式不变量的显式基础（与Etsuko Bannal， Michio Ozeki和Yasuo Teranishi联合工作）。（iv）我们目前正在研究有限阿贝群上的II型加性码（与Masaaki Harada， s.d aherty和Manabu Oura合作）。利用正多面体和拟正多面体的分类等初等几何思想，对m_1=3的原始对称关联方案进行了分类。我目前正在与Attila Sali合作研究m_1=4的原始对称q -多项式关联方案的分类。

The head researcher has done the research in the following 3 directions.1.Study of spin models. (i) We introduced the concept of 4-weight spin models, and proved that link invariants are obtained from them (joint work with Etsuko Bannai). (ii) We gave the complete classifications of the dualities and the modular invariances on the character tables of finite abelian groups, and proved that spin models are constructed from such solutions (joint work with Etsuko Bannai and F.Jaeger).2.Construction of various automorphic forms from either the weight enumerators of codes or polynomial invariants of certain finite groups.(i) We proved that Jacobi forms are constructed from the simultaneous diagonal actions of the 2-dimensional unitary reflection group of order 192 (No.9 in Shepherd-Todd's list)(joint work with Michio Ozeki).(ii) Explicit constructions of certain Jacobi forms of weight 4 (joint work with Michio Ozeki and Shinri Minashima).(iii) We determined the explicit basis of the polynomial invariants mentioned in (i) above (joint work with Etsuko Bannal, Michio Ozeki and Yasuo Teranishi).(iv) We are currently studying Type II additive codes on finite abelian groups (joint work with Masaaki Harada, S.Dougherty, and Manabu Oura).3.I classified primitive symmetric association schemes with m_1=3 by using elementary geometric ideas such as the classifications of regular polyhedrons and quasi-regular polyhedrons. I am currently working on the classification of primitive symmetric Q-polynomial association schemes with m_1=4 jointly with Attila Sali.